Very hard to do, especially because you are unlikely to have a suitable US 'credit score' without adequate presence/address/job/social security number there. I have a US SSN, suitable non-immigrant residency (L1) working visa, and substantive business operations/banking relationships in the US. My US bank suggested they might be able to pull strings to get a credit card, but could I live with a Visa debit card instead? So that 'works' for me -- but no true US credit card even now.
I don't know about this specific card, but I am sure the rules may vary.
I have two different US-issued credit cards. One with a US PO-box address (to deal with those pesky people who will only take payment with credit cards with US billing addresses), the second is with my old credit unit and they use my Canadian mailing address. I could have easily had both credit cards use my Canadian mailing / billing address. Living only a few kilometres from the border, the PO box is no big deal. Living much further north, there might be a service (e.g., MailBoxEtc) that might forward the contents of a PO box for a fee.
I don't believe there is any general rule - it is up to the company issuing the card whether to impose a residency requirement. Many Canadians who live near the border or winter in Florida have JC Penney cards, for example. Marriott prefers a US address for some reason.
I too have always wanted the Marriott card, as it is one of the more difficult programs to accumulate points in without one.
Not the yyzflyer on Airliners.net (in case there was any doubt)